Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.642 - 0.766i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.863·5-s i·7-s − 2.62i·11-s + 1.01i·13-s + 2.34i·17-s + 4.09·19-s − 6.23·23-s − 4.25·25-s − 1.57·29-s + 6.29i·31-s − 0.863i·35-s + 5.18i·37-s + 10.1i·41-s + 0.496·43-s + 5.24·47-s + ⋯
L(s)  = 1  + 0.386·5-s − 0.377i·7-s − 0.792i·11-s + 0.282i·13-s + 0.568i·17-s + 0.939·19-s − 1.29·23-s − 0.850·25-s − 0.292·29-s + 1.13i·31-s − 0.145i·35-s + 0.851i·37-s + 1.58i·41-s + 0.0756·43-s + 0.764·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.642 - 0.766i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.642 - 0.766i)$
$L(1)$  $\approx$  $1.777017086$
$L(\frac12)$  $\approx$  $1.777017086$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 0.863T + 5T^{2} \)
11 \( 1 + 2.62iT - 11T^{2} \)
13 \( 1 - 1.01iT - 13T^{2} \)
17 \( 1 - 2.34iT - 17T^{2} \)
19 \( 1 - 4.09T + 19T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 + 1.57T + 29T^{2} \)
31 \( 1 - 6.29iT - 31T^{2} \)
37 \( 1 - 5.18iT - 37T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 - 0.496T + 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 4.40iT - 59T^{2} \)
61 \( 1 - 9.59iT - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 2.11T + 71T^{2} \)
73 \( 1 + 8.27T + 73T^{2} \)
79 \( 1 + 9.44iT - 79T^{2} \)
83 \( 1 + 6.62iT - 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.171880654502782434873991883791, −7.51436981589557301615314904122, −6.70950013810355448702255711563, −5.97150798395764124879442024340, −5.49077671873469782056080246775, −4.47714020174526426409491514365, −3.74560018851137682617249887196, −2.97019361715637716924020912573, −1.90402957536163767822919006187, −0.985446495321195657642635452184, 0.50493093264385298419005431003, 1.95698808706350335899055524786, 2.40491690932448481142113431272, 3.62069621996836836485742398539, 4.25096771107005361097664768147, 5.39452907317464475569045203361, 5.61394715305397820163782311334, 6.55288102826981611813942273197, 7.41086911614151389499606566166, 7.80268577880323732054664820597

Graph of the $Z$-function along the critical line